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JIMO

In this paper, we discuss a system of differential equations based
on the projection operator for solving the box constrained
variational inequality problems. The equilibrium solutions to the
differential equation system are proved to be the solutions of the
box constrained variational inequality problems. Two differential
inclusion problems associated with the system of differential
equations are introduced. It is proved that the equilibrium
solution to the differential equation system is locally
asymptotically stable by verifying the locally asymptotical
stability of the equilibrium positions of the differential
inclusion problems. An Euler discrete scheme with Armijo line
search rule is introduced and its global convergence is
demonstrated. The numerical experiments are reported to show that
the Euler method is effective.

DCDS

In this paper, we consider the following problem
$$
\left\{
\begin{array}{ll}
-\Delta u+u=u^{2^{*}-1}+\lambda(f(x,u)+h(x))\ \ \hbox{in}\ \mathbb{R}^{N},\\
u\in H^{1}(\mathbb{R}^{N}),\ \ u>0 \ \hbox{in}\ \mathbb{R}^{N},
\end{array}
\right. (\star)
$$
where $\lambda>0$ is a parameter, $2^* =\frac {2N}{N-2}$ is the critical Sobolev exponent and $N>4$, $f(x,t)$ and $h(x)$ are some given functions. We
prove that there exists $0<\lambda^{*}<+\infty$ such that $(\star)$ has
exactly two positive solutions for $\lambda\in(0,\lambda^{*})$ by
Barrier method and Mountain Pass Lemma and no positive solutions for $\lambda >\lambda^*$. Moreover,
if $\lambda=\lambda^*$, $(\star)$ has a unique solution $(\lambda^{*}, u_{\lambda^{*}})$, which means that $(\lambda^{*}, u_{\lambda^{*}})$ is a
turning point in $H^{1}(\mathbb{R}^{N})$ for problem $(\star)$.

DCDS

In this paper, by an approximating argument, we obtain infinitely many radial solutions for the
following elliptic systems with critical Sobolev growth
$$
\left\lbrace\begin{array}{l}
-\Delta u=|u|^{2^*-2}u +
\frac{η \alpha}{\alpha+β}|u|^{\alpha-2}u |v|^β + \frac{σ p}{p+q} |u|^{p-2}u|v|^q , \ \ x ∈ B , \\
-\Delta v = |v|^{2^*-2}v + \frac{η β}{\alpha+ β } |u|^{\alpha }|v|^{β-2}v
+ \frac{σ q}{p+q} |u|^{p}|v|^{q-2}v , \ \ x ∈ B , \\
u = v = 0, \ \ &x \in \partial B, \end{array}\right.
$$
where $N > \frac{2(p + q + 1) }{p + q - 1}, η, σ > 0, \alpha,β > 1$ and $\alpha + β = 2^* = : \frac{2N}{N-2} ,$ $p,\,q\ge 1$, $2\le p +q<2^*$ and $B\subset \mathbb{R}^N$ is an open ball centered at the origin.

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